Question

Which of these graphs shows
that the linear
system 3x+2y=12 and y=-3/2x+6
has an infinite number of
solutions? (1 point)

Answer 1

To determine if the system of equations \(3x + 2y = 12\) and \(y = -\frac{3}{2}x + 6\) has an infinite number of solutions, we need to check if the two equations represent the same line.

1. The first equation \(3x + 2y = 12\) can be rewritten in slope-intercept form (y = mx + b):
\[
2y = -3x + 12
\]
\[
y = -\frac{3}{2}x + 6
\]
This form shows that the line's slope is \(-\frac{3}{2}\) and the y-intercept is \(6\).

2. The second equation is already in slope-intercept form:
\[
y = -\frac{3}{2}x + 6
\]

Since both equations represent the same line (\(y = -\frac{3}{2}x + 6\)), this means they have an infinite number of solutions.

To find the graph that shows this, look for a graph where both lines overlap completely. In other words, both equations graph as the same line on the graph coordinate system. The correct graph would ideally show just one line rather than two distinct lines.

If you have multiple graph options, select the one where the two equations overlap perfectly.